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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

A separator theorem for nonplanar graphs


Authors: Noga Alon, Paul Seymour and Robin Thomas
Journal: J. Amer. Math. Soc. 3 (1990), 801-808
MSC: Primary 05C40; Secondary 05C85, 68Q25, 68R10
MathSciNet review: 1065053
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Abstract: Let $ G$ be an $ n$-vertex graph with no minor isomorphic to an $ h$-vertex complete graph. We prove that the vertices of $ G$ can be partitioned into three sets $ A,\;B,\;C$ such that no edge joins a vertex in $ A$ with a vertex in $ B$, neither $ A$ nor $ B$ contains more than $ 2n/3$ vertices, and $ C$ contains no more than $ {h^{3/2}}{n^{1/2}}$ vertices. This extends a theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds such a partition $ (A,\;B,\;C)$ in time $ O({h^{1/2}}{n^{1/2}}m)$, where $ m = \left\vert {V(G)} \right\vert + \left\vert {E(G)} \right\vert$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0
PII: S 0894-0347(1990)1065053-0
Article copyright: © Copyright 1990 American Mathematical Society